RD SHARMA SOLUTION CHAPTER –20 Mensuration – I | CLASS 7TH MATHEMATICS-EDUGROWN
Exercise 20.1
Question: 1
Find the area, in square meters, of a rectangle whose
(i) Length = 5.5 m, breadth = 2.4 m
(ii) Length = 180 cm, breadth = 150 cm
Solution:
We have,
(i) Length = 5.5 m, Breadth = 2.4 m Therefore, Area of rectangle = Length x Breadth = 5.5 m x 2.4 m = 13.2 m2
(ii) Length = 180 cm = 1.8 m, Breadth = 150 cm = 1.5 m [ Since 100 cm = 1 m] Therefore, Area of rectangle = Length x Breadth = 1.8 m x 1.5 m = 2.7 m2
Question: 2
Find the area, in square centimeters, of a square whose side is
(i) 2.6 cm
(ii) 1.2 dm
Solution:
We have,
(i) Side of the square = 2.6 cm
Therefore, area of the square = (Side)2 = (2.6 cm)2= 6.76 cm2
(ii) Side of the square = 1.2 dm = 1.2 x 10 cm = 12 cm
Therefore, area of the square = (Side)2 = (12 cm)2= 144 cm2 [ Since 1 dm = 10 cm]
Question: 3
Find in square metres, the area of a square of side 16.5 dam.
Solution:
We have,
Side of the square = 16.5
dam = 16.5 x 10 m = 165 m
Area of the square = (Side)2 = (165 m)2 = 27225 m2
[Since 1 dam/dm (decameter) = 10 m]
Question: 4
Find the area of a rectangular field in acres whose sides are:
(1) 200 m and 125 m
(ii) 75 m 5 dm and 120 m
Solution:
We have,
(i) Length of the rectangular field = 200 m
Breadth of the rectangular field = 125 m
Therefore, Area of the rectangular field = Length x Breadth = 200 m x 125 m
= 25000 m2 = 250 acres [Since 100 m2 = 1 are]
(ii) Length of the rectangular field =75 m 5 dm = (75 + 0.5) m
= 75.5 m [Since 1 dm = 10 cm = OA m]
Breadth of the rectangular field = 120 m
Therefore, Area of the rectangular field = Length x Breadth
= 75.5 m x 120 m = 9060 m2 = 90.6 acres [Since 100 m2 = 1 are]
Question: 5
Find the area of a rectangular field in hectares whose sides are:
(i) 125 m and 400 m
(ii) 75 m 5 dm and 120 m
Solution:
We have,
(i) Length of the rectangular field = 125 m
Breadth of the rectangular field = 400 m
Therefore, Area of the rectangular field = Length x Breadth
= 125 m x 400 m = 50000 m2 = 5 hectares [Since 10000 m2 = 1 hectare]
(ii) Length of the rectangular field =75 m 5 dm = (75 + 0.5) m
= 75.5 m [Since 1 dm = 10 cm = 0.1 m]
Breadth of the rectangular field = 120 m
Therefore, Area of the rectangular field = Length x Breadth
= 75.5 m x 120 m = 9060 m2 = 0.906 hectares [Since 10000 m2 = 1 hectare]
Question: 6
A door of dimensions 3 m x 2m is on the wall of dimension 10 m x 10 m. Find the cost of painting the wall if rate of painting is Rs 2.50 per sq. m.
Solution:
We have,
Length of the door = 3 m
Breadth of the door = 2 m
Side of the wall = 10 m
Area of the wall = Side x Side = 10 m x 10 m
= 100 m2
Area of the door = Length x Breadth = 3 m x 2 m = 6 m
Thus, required area of the wall for painting = Area of the wall – Area of the door
= (100 – 6) m2= 94 m2
Rate of painting per square metre = Rs. 2.50
Hence, the cost of painting the wall = Rs. (94 x 2.50) = Rs. 235
Question: 7
A wire is in the shape of a rectangle. Its length is 40 cm and breadth is 22 cm. If the same wire is bent in the shape of a square, what will be the measure of each side? Also, find which side encloses more area?
Solution:
We have,
Perimeter of the rectangle = 2(Length + Breadth)
= 2(40 cm + 22 cm) = 124 cm
It is given that the wire which was in the shape of a rectangle is now bent into a square.
Therefore, the perimeter of the square = Perimeter of the rectangle
→ Perimeter of the square = 124 cm
4 x side = 124 cm
Side = 124/4 = 31 cm
Now, Area of the rectangle = 40 cm x 22 cm = 880 cm2
Area of the square = (Side)2 = (31 cm)2 = 961 cm2.
Therefore, the square-shaped wire encloses more area.
Question: 8
How many square metres of glass will be required for a window, which has 12 panes, each pane measuring 25 cm by 16 cm?
Solution:
We have,
Length of the glass pane = 25 cm
Breadth of the glass pane = 16 cm
Area of one glass pane = 25 cm x 16 cm
= 400 cm2 = 0.04 m2
[Since 1 m2 = 10000 cm2 ]
Thus, Area of 12 such panes = 12 x 0.04 = 0.48 m2
Question: 9
A marble tile measures 10 cm x 12 cm. How many tiles will be required to cover a wall of size 3 m x 4 m? Also, find the total cost of the tiles at the rate of Rs 2 per tile.
Solution:
We have,
Area of the wall = 3 m x 4 m = 12 m2
Area of one marble tile = 10 cm x 12 cm
= 120 c m2 = 0.012 m2 [Since 1 m2 = 10000 c m2 ]
Thus, Number of tiles = Area of wall
Area of one tile=12 m2 = 0.012 m2=1000
Cost of one tile = Rs. 2
Total cost = Number of tiles x Cost of one tile
= Rs. (1000 x 2)
= Rs. 2000
Question: 10
A table top is 9 dm 5 cm long 6 dm 5 cm broad. What will be the cost to polish it at the rate of 20 paise per square centimetre?
Solution:
We have,
Length of the table top = 9 dm 5 cm = (9 x 10 + 5) cm = 95 cm [ Since 1 dm = 10 cm]
Breadth of the table top = 6 dm 5 cm = (6 x 10 + 5) cm = 65 cm
Area of the table top = Length x Breadth = (95 cm x 65 cm) = 6175 c m2
Rate of polishing per square centimetre = 20 paise = Rs. 0.20
Total cost = Rs. (6175 x 0.20) = Rs. 1235
Question: 11
A room is 9.68 m long and 6.2 m wide. Its floor is to be covered with rectangular tiles of size 22 cm by 10 cm. Find the total cost of the tiles at the rate of Rs 2.50 per tile.
Solution:
We have,
Length of the floor of the room = 9.68 m
Breadth of the floor of the room = 6.2 m
Area of the floor = 9.68 m x 6.2 m = 60.016 m2
Length of the tile = 22 cm
Breadth of the tile = 10 cm
Area of one tile = 22 cm x 10 cm = 220 c m2 = 0.022 m2 [Since 1 m2 = 10000 c m2]
Thus, Number of tiles = 60.016 m2/0.022 m2=2728
Cost of one tile = Rs. 2.50
Total cost = Number of tiles x Cost of one tile = Rs. (2728 x 2.50) = Rs. 6820
Question: 12
One side of a square field is 179 m. Find the cost of raising a lawn on the field at the rate of Rs 1.50 per square metre.
Solution:
We have,
Side of the square field = 179 m
Area of the field = (Side) 2 = (179 m) 2 = 32041 m2
Rate of raising a lawn on the field per square metre = Rs. 1.50 Thus,
Total cost of raising a lawn on the field = Rs. (32041 x 1.50) = Rs. 48061.50
Question: 13
A rectangular field is measured 290 m by 210 m. How long will it take for a girl to go two times round the field, if she walks at the rate of 1.5 m/sec?
Solution:
We have,
Length of the rectangular field = 290 m
Breadth of the rectangular field = 210 m
Perimeter of the rectangular field = 2(Length + Breadth) = 2(290 + 210) = 1000 m
Distance covered by the girl = 2 x Perimeter of the rectangular field = 2 x 1000 = 2000 m
The girl walks at the rate of 1.5 m/sec. Or, Rate = 1.5 x 60 m/min = 90 m/min
Thus, required time to cover a distance of 2000 m = 2000 m/90 m/min
= 2229min
Hence, the girl will take 2229 min to go two times around the field.
Question: 14
A corridor of a school is 8 m long and 6 m wide. It is to be covered with canvas sheets. If the available canvas sheets have the size 2 m x 1 m, find the cost of canvas sheets required to cover the corridor at the rate of Rs 8 per sheet.
Solution:
We have,
Length of the corridor = 8 m
Breadth of the corridor = 6 m
Area of the corridor of a school = Length x Breadth = (8 m x 6 m) = 48 m2
Length of the canvas sheet = 2 m
Breadth of the canvas sheet = 1 m
Area of one canvas sheet = Length x Breadth = (2 m x 1 m) = 2 m2
Thus, Number of canvas sheets = 48 m2 /2m2=24
Cost of one canvas sheet = Rs. 8
Total cost of the canvas sheets = Rs. (24 x 8) = Rs. 192
Question: 15
The length and breadth of a playground are 62 m 60 cm and 25 m 40 cm respectively. Find the cost of turfing it at Rs 2.50 per square metre. How long will a man take to go three times round the field, if he walks at the rate of 2 metres per second?
Solution:
We have,
Length of a playground = 62 m 60 cm = 62.6 m [ Since 10 cm = 0.1 m]
Breadth of a playground = 25 m 40 cm = 25.4 m
Area of a playground = Length x Breadth= 62.6 m x 25.4 m = 1590.04 m2
Rate of turfing = Rs. 2.50/ m2 Total cost of turfing = Rs. (1590.04 x 2.50) = Rs. 3975.10
Again, Perimeter of a rectangular field = 2(Length + Breadth) = 2(62.6 + 25.4) = 176 m
Distance covered by the man in 3 rounds of a field = 3 x Perimeter of a rectangular field
= 3 x 176 m = 528 m
The man walks at the rate of 2 m/sec. Or, Rate = 2 x 60 m/min = 120 m/min
Thus, required time to cover a distance of 528 m = 528 m120 m/min=4.4 min
= 4 minutes 24 seconds [Since 0.1 minutes = 6 seconds]
Question: 16
A lane 180 m long and 5 m wide is to be paved with bricks of length 20 cm and breadth 15 cm. Find the cost of bricks that are required, at the rate of Rs 750 per thousand.
Solution:
We have,
Length of the lane = 180 m
Breadth of the lane = 5 m
Area of a lane = Length x Breadth = 180 m x 5 m = 900 m2
Length of the brick = 20 cm
Breadth of the brick = 15 cm
Area of a brick = Length x Breadth = 20 cm x 15 cm
= 300 cm2 = 0.03 m2 [Since 1 m2 = 10000 cm2]
Required number of bricks = 900 m2/0.03 m2 = 30000
Cost of 1000 bricks = Rs. 750
Total cost of 30,000 bricks = Rs. 750×30,000/1000 = Rs. 22,500
Question: 17
How many envelopes can be made out of a sheet of paper 125 cm by 85 cm; supposing one envelope requires a piece of paper of size 17 cm by 5 cm?
Solution:
We have,
Length of the sheet of paper = 125 cm
Breadth of the sheet of paper = 85 cm
Area of a sheet of paper = Length x Breadth = 125 cm x 85 cm = 10,625 cm2
Length of sheet required for an envelope = 17 cm
Breadth of sheet required for an envelope = 5 cm
Area of the sheet required for one envelope = Length x Breadth
= 17 cm x 5 cm = 85 c m2
Thus, required number of envelopes = 10,625 cm2/85 c m2 = 125
Question: 18
The width of a cloth is 170 cm. Calculate the length of the cloth required to make 25 diapers, if each diaper requires a piece of cloth of size 50 cm by 17 cm.
Solution:
We have,
Length of the diaper = 50 cm
Breadth of the diaper = 17 cm
Area of cloth to make 1 diaper = Length x Breadth = 50 cm x 17 cm = 850 cm2
Thus, Area of 25 such diapers = (25 x 850) c m2 = 21,250 cm2
Area of total cloth = Area of 25 diapers = 21,250 cm2
It is given that width of a cloth = 170 cm
Length of the cloth = Area of cloth
Width of a cloth = 21,250 cm 2170 cm = 125 cm
Hence, length of the cloth will be 125 cm.
Question: 19
The carpet for a room 6.6 m by 5.6 m costs Rs 3960 and it was made from a roll 70 cm wide. Find the cost of the carpet per metre.
Solution:
We have,
Length of a room = 6.6 m
Breadth of a room = 5.6 m
Area of a room = Length x Breadth = 6.6 m x 5.6 m = 36.96 m2
Width of a carpet = 70 cm = 0.7 m [Since 1 m = 100 cm]
Length of a carpet = Area of a room
Width of a carpet = 36.96 m 20.7 m = 52.8 m
Cost of 52.8 m long roll of carpet = Rs. 3960
Therefore, Cost of 1 m long roll of carpet = Rs. 396052.8 = Rs. 75
Question: 20
A room is 9 m long, 8 m broad and 6.5 m high. It has one door of dimensions 2 m x 1.5 m and three windows each of dimensions 1.5 m x 1 m. Find the cost of white T4L washing the walls at Rs 3.80 per square metre.
Solution:
We have,
Length of a room = 9 m
Breadth of a room = 8 m
Height of a room = 6.5 m
Area of 4 walls = 2(1 + b)h = 2(9 m + 8 m) x 6.5 m = 2 x 17 m x 6.5 m = 221 m2
Length of a door = 2 m
Breadth of a door = 1.5 m
Area of a door = Length x Breadth = 2 m x 1.5 m = 3 m2
Length of a window = 1.5 m
Breadth of a window = 1 m
Since, area of one window = Length x Breadth = 1.5 m x 1 m = 1.5 m2
Thus, Area of 3 such windows = 3 x 1.5 m2 = 4.5 m2
Area to be white-washed = Area of 4 walls – (Area of one door + Area of 3 windows)
Area to be white-washed = [221 – (3 + 4.5)] m2 = (221 – 7.5) m2 = 213.5 m2
Cost of white-washing for 1 m2 area = Rs. 3.80
Cost of white-washing for 213.5 m2 area = Rs. (213.5 x 3.80) = Rs. 811.30
Question: 21
A hall 36 m long and 24 m broad allowing 80 m2 for doors and windows, the cost of papering the walls at Rs 8.40 per m2 is Rs 9408. Find the height of the hall.
Solution:
We have,
Length of the hall = 36 m
Breadth of the hall = 24 m
Let h be the height of the hall.
Now, in papering the wall, we need to paper the four walls excluding the floor and roof of the hall. So, the area of the wall which is to be papered = Area of 4 walls
= 2h(I + b)
= 2h (36 + 24)
= 120h m2
Now, area left for the door and the windows = 80 m2
So, the area which is actually papered = (120h – 80) m2
Again, The cost of papering the walls at Rs 8.40 per m2 = Rs. 9408
→ (120h – 80) m2 x Rs. 8.40 per m2= Rs. 9408
→ (120h – 80) m2 = Rs. 9408/Rs. 8.40
→ (120h – 80) m2 = 1120 m2
→ 120h m2 = (1120 + 80) m2
→ 120h m2= 1200 m2
h = 1200 m2 120 m = 10 m
Hence, the height of the wall would be 10 m.
Exercise 20.2
Question: 1
A rectangular grassy lawn measuring 40 m by 25 m is to be surrounded externally by a path which is 2 m wide. Calculate the cost of leveling the path at the rate of Rs 8.25 per square metre.
Solution:
We have,
Length AB = 40 m and breadth BC = 25 m
Area of lawn ABCD = 40 m x 25 m = 1000 m2
Length PQ = (40 + 2 + 2 ) m = 44 m
Breadth QR = ( 25 + 2 + 2 ) m = 29 m
Area of PQRS = 44 m x 29 m = 1276 m2
Now, Area of the path = Area of PQRS – Area of the lawn ABCD
= 1276 m2 – 1000 m2 = 276 m2
Rate of leveling the path = Rs. 8.25 per m2
Cost of leveling the path = Rs.( 8.25 x 276) = Rs. 2277
Question: 2
One metre wide path is built inside a square park of side 30 m along its sides. The remaining part of the park is covered by grass. If the total cost of covering by grass is Rs 1176, find the rate per square metre at which the park is covered by the grass.
Solution:
We have,
Side of square garden (a) = 30 m
Area of the square garden including the path = a2= (30)2 = 900 m2
From the figure, it can be observed that the side of the square garden, when the path is not included, is 28 m.
Area of the square garden not including the path = (28)2 = 784 m2
Total cost of covering the park with grass = Area of the park covering with green grass x Rate per square metre
1176 = 784 x Rate per square metre
Rate per square metre at which the park is covered with grass = Rs. (1176 ÷ 784 ) = Rs. 1.50
Question: 3
Through a rectangular field of sides 90 m x 60 m, two roads are constructed which are parallel to the sides and cut each other at right angles through the centre of the field. if the width of the roads is 3 m, find the total area covered by the two roads.
Solution:
Length of the rectangular sheet = 90 m
Breadth of the rectangular sheet = 60 cm
Area of the rectangular field = 90 m x 60 m = 5400 m2
Area of the road PQRS = 90 m x 3 m = 270 m2
Area of the road ABCD = 60 m x 3 m = 180 m2
Clearly, area of KLMN is common to the two roads.
Thus, area of KLMN = 3 m x 3 m = 9 m2
Hence, Area of the roads = Area (PQRS) + Area (ABCD) – Area (KLMN)
= (270 + 180) m2 – 9 m2 = 441 m2
Question: 4
from a rectangular sheet of tin, of size 100 cm by 80 cm, are cut four squares of side 10 cm from each corner. Find the area of the remaining sheet.
Solution:
Length of the rectangular sheet = 100 cm
Breadth of the rectangular sheet = 80 cm
Area of the rectangular sheet of tin = 100 cm x 80 cm
= 8000 c m2
Side of the square at the corner of the sheet = 10 cm
Area of one square at the corner of the sheet = (10 cm)2 = 100 cm2
Area of 4 squares at the corner of the sheet = 4 x 100 cm2 = 400 cm2
Hence, Area of the remaining sheet of tin = Area of the rectangular sheet – Area of the 4 squares
Area of the remaining sheet of tin = (8000 – 400) cm2 = 7600 cm2
Question: 5
A painting 8 cm long and 5 cm wide is painted on a cardboard such that there is a margin of 1.5 cm along each of its sides. Find the total area of the margin.
Solution:
We have, Length of the cardboard = 8 cm and breadth of the cardboard = 5 cm
Area of the cardboard including the margin = 8 cm x 5 cm = 40 c m2
From the figure, it can be observed that,
New length of the painting when the margin is not included = 8 cm – (1.5 cm + 1.5 cm)
= (8 – 3) cm = 5 cm
New breadth of the painting when the margin is not included = 5 cm – (1.5 cm + 1.5 cm)
= (5 – 3) cm = 2 cm
Area of the painting not including the margin = 5 cm x 2 cm = 10 cm2
Hence, Area of the margin = Area of the cardboard including the margin – Area of the painting
= (40 – 10) cm2 = 30 cm2
Question: 6
Rakesh has a rectangular field of length 80 m and breadth 60 m. In it, he wants to make a garden 10 m long and 4 m broad at one of the corners and at another corner, he wants to grow flowers in two floor-beds each of size 4 m by 1.5 m. In the remaining part of the field, he wants to apply manures. Find the cost of applying the manures at the rate of Rs 300 per area.
Solution:
Length of the rectangular field = 80 m
Breadth of the rectangular field = 60 m
Area of the rectangular field = 80 m x 60 = 4800 m2
Again, Area of the garden = 10 m x 4 m = 40 m2
Area of one flower bed = 4 m x 1.5 m = 6 m2
Thus, Area of two flower beds = 2 x 6 m2 = 12 m2
Remaining area of the field for applying manure = Area of the rectangular field – (Area of the garden + Area of the two flower beds)
Remaining area of the field for applying manure = 4800 m2 – (40 + 12) m2
= (4800 – 52 ) m2 = 4748 m2
Since 100 m2 = 1 acre → 4748 m2 = 47.48 acres
So, cost of applying manure at the rate of Rs. 300 per are will be Rs. (300 x 47.48) = Rs. 14244
Question: 7
Each side of a square flower bed is 2 m 80 cm long. It is extended by digging a strip 30 cm wide all around it. Find the area of the enlarged flower bed and also the increase in the area of the flower bed.
Solution:
We have ,
Side of the flower bed = 2 m 80 cm = 2.80 m [since 100 cm = 1 m ]
Area of the square flower bed = (Side)2 = (2.80 m )2 = 7.84 m2
Side of the flower bed with the digging strip = 2.80 m + 30 cm + 30 cm
= (2.80 + 0.3 + 0.3) m
= 3.4 m
Area of the enlarged flower bed with the digging strip = (Side) 2 = (3.4)2 = 11.56 m2
Thus, Increase in the area of the flower bed = 11.56 m2 – 7.84 m2 = 3.72 m2
Question: 8
A room 5 m long and 4 m wide is surrounded by a verandah. If the verandah occupies an area of 22 m2 , find the width of the varandah.
Solution:
Let the width of the verandah be x m.
Length of the room AB = 5 m and BC = 4 m
Area of the room = 5 m x 4 m = 20 m2
Length of the verandah PQ = (5 + x + x) = (5 + 2x) m
Breadth of the verandah QR = ( 4 + x + x) = (4 + 2x) m
Area of verandah PQRS = (5 + 2x) x (4 + 2x) = (4×2 + 18x + 20 ) m2
Area of verandah = Area of PQRS – Area of ABCD
→ 22 = 4x2 + 18x + 20 – 20
22 = 4x2 + 18x
11 = 2x2 + 9x
2x2 + 9x – 11 = 0
2x2 + 11x – 2x – 11 =0
x(2x+11)-1(2x+11)=0
(x- 1)(2x+11)= 0
When x – 1 = 0, x = 1
When 2x + 11 = 0, x = -11/2
The width cannot be a negative value. So, width of the verandah = x = 1 m.
Question: 9
A square lawn has a 2 m wide path surrounding it. If the area of the path is 136 m2 , find the area of the lawn.
Solution:
Let ABCD be the square lawn and PQRS be the outer boundary of the square path.
Let side of the lawn AB be x m.
Area of the square lawn = x2
Length PQ = (x m + 2 m + 2 m) = (x + 4) m
Area of PQRS = (x + 4)2 = (x2 + 8x + 16) m2
Now, Area of the path = Area of PQRS – Area of the square lawn
136 = x2 + 8x + 16 – x2
136 = 8x + 16
136 – 16 = 8x
120 = 8x
x = 120/ 8 = 15
Side of the lawn = 15 m Hence, Area of the lawn = (Side)2 = (15 m)2 = 225 m2
Question: 10
A poster of size 10 cm by 8 cm is pasted on a sheet of cardboard such that there is a margin of width 1.75 cm along each side of the poster. Find (i) the total area of the margin (ii) the cost of the cardboard used at the rate of Re 0.60 per c m2 .
Solution:
We have,
Length of poster = 10 cm and breadth of poster = 8 cm
Area of the poster = Length x Breadth = 10 cm x 8 cm = 80 cm2
From the figure, it can be observed that,
Length of the cardboard when the margin is included = 10 cm + 1.75 cm + 1.75 cm = 13.5 cm
Breadth of the cardboard when the margin is included = 8 cm + 1.75 cm + 1.75 cm = 11.5 cm
Area of the cardboard = Length x Breadth = 13.5 cm x 11.5 cm = 155.25 c m2
Hence,
(i) Area of the margin = Area of cardboard including the margin – Area of the poster
= 155.25 c m2 – 80 c m2
= 75.25 c m2
(ii) Cost of the cardboard = Area of cardboard x Rate of the cardboard Rs 0.60 per cm2
= Rs. (155.25 x 0.60)
= Rs. 93.15
Question: 11
A rectangular field is 50 m by 40 m. It has two roads through its centre, running parallel to its sides. The widths of the longer and shorter roads are 1.8 m and 2.5 m respectively. Find the area of the roads and the area of the remaining portion of the field.
Solution:
Let ABCD be the rectangular field and KLMN and PQRS the two rectangular roads with width 1.8 m and 2.5 m, respectively.
Length of the rectangular field CD = 50 cm and breadth of the rectangular field BC = 40 m
Area of the rectangular field ABCD = 50 m x 40 m = 2000 m2
Area of the road KLMN = 40 m x 2.5 m = 100 m2
Area of the road PQRS = 50 m x 1.8 m = 90 m2
Clearly area of EFGH is common to the two roads.
Thus, Area of EFGH = 2.5 m x 1.8 m = 4.5 m2
Hence, Area of the roads = Area (KLMN) + Area (PQRS) – Area (EFGH)
= (100 m2 + 90 m2) – 4.5 m2
= 185.5 m2
Area of the remaining portion of the field = Area of the rectangular field ABCD – Area of the roads
= (2000 – 185.5) m2
= 1814.5 m2
Question: 12
There is a rectangular field of size 94 m x 32 m. Three roads each of 2 m width pass through the field such that two roads are parallel to the breadth of the field and the third is parallel to the length. Calculate: (i) area of the field covered by the three roads (ii) area of the field not covered by the roads.
Solution:
Let ABCD be the rectangular field.
Here, Two roads which are parallel to the breadth of the field KLMN and EFGH with width 2 m each. One road which is parallel to the length of the field PQRS with width 2 m.
Length of the rectangular field AB = 94 m and breadth of the rectangular field BC = 32 m
Area of the rectangular field = Length x Breadth = 94 m x 32 m = 3008 m2
Area of the road KLMN = 32 m x 2 m = 64 m2
Area of the road EFGH = 32 m x 2 m = 64 m2
Area of the road PQRS = 94 m x 2 m = 188 m2
Clearly area of TUVI and WXYZ is common to these three roads.
Thus, Area of TUV1 = 2 m x 2 m = 4 m2
Area of WXYZ = 2 m x 2 m = 4 m2
Hence,
(i) Area of the field covered by the three roads: = Area (KLMN) + Area (EFGH) + Area (PQRS) – {Area (TUVI) + Area (WXYZ)}
= [ 64+ 64 + 188 – (4 + 4 )] m2
= 316 m2 – 8 m2
= 308 m2
(ii) Area of the field not covered by the roads: = Area of the rectangular field ABCD – Area of the field covered by the three roads
= 3008 m2 – 308 m2
= 2700 m2
Question: 13
A school has a hall which is 22 m long and 15.5 m broad. A carpet is laid inside the hall leaving all around a margin of 75 cm from the walls. Find the area of the carpet and the area of the strip left uncovered. If the width of the carpet is 82 cm, find the cost at the rate of Rs 18 per metre.
Solution:
We have,
Length of hall PQ = 22 m and breadth of hall QR = 15.5 m
Area of the school hall PQRS = 22 m x 15.5 m = 341 m2
Length of the carpet AB = 22 m – ( 0.75 m + 0.75 m) = 20.5 m [ Since 100 cm = 1 m]
Breadth of the carpet BC = 15.5 m – ( 0.75 m + 0.75 m) = 14 m
Area of the carpet ABCD = 20.5 m x 14 m = 287 m2
Area of the strip = Area of the school hall PQRS – Area of the carpet ABCD
= 341 m2 – 287 m2 = 54 m2
Again, Area of the 1 m length of carpet = 1 m x 0.82 m = 0.82 m2
Thus, Length of the carpet whose area is 287 m2 = 287 m2 + 0.82 m2 = 350 m
Cost of the 350 m long carpet = Rs. 18 x 350 = Rs. 6300
Question: 14
Two cross roads, each of width 5 m, run at right angles through the centre of a rectangular park of length 70 m and breadth 45 m parallel to its sides. Find the area of the roads. Also, find the cost of constructing the roads at the rate of Rs 105 per m2 .
Solution:
Let ABCD be the rectangular park then EFGH and IJKL the two rectangular roads with width 5 m.
Length of the rectangular park AD = 70 cm
Breadth of the rectangular park CD = 45 m
Area of the rectangular park = Length x Breadth = 70 m x 45 m = 3150 m2
Area of the road EFGH = 70 m x 5 m = 350 m2
Area of the road JKIL = 45 m x 5 m = 225 m2
Clearly area of MNOP is common to the two roads.
Thus, Area of MNOP = 5 m x 5 m = 25 m2
Hence,
Area of the roads = Area (EFGH) + Area (JKIL) – Area (MNOP)
= (350 + 225) m2– 25 m2 = 550 m2
Again, it is given that the cost of constructing the roads = Rs. 105 per m2
Therefore,
Cost of constructing 550 m2 area of the roads = Rs. (105 x 550)
= Rs. 57750.
Question: 15
The length and breadth of a rectangular park are in the ratio 5: 2. A 2.5 m wide path running all around the outside the park has an area 305 m2 . Find the dimensions of the park.
Solution:
We have,
Area of path = 305 m2
Let the length of the park be 5x m and the breadth of the park be 2x m
Thus,
Area of the rectangular park = (5x) x (2x) = 10x2 m2
Width of the path = 2.5 m
Outer length PQ = 5x m + 2.5 m + 2.5 m = (5x + 5) m
Outer breadth QR = 2x + 2.5 m + 2.5 m = (2x + 5) m
Area of PQRS = (5x + 5) m x (2x + 5) m = (10x2 + 25x + 10x + 25) m2= (10x2 + 35x + 25) m2
Area of the path = [(10x2 + 35x + 25) – 10x2] m2
→ 305 = 35x + 25
→ 305 – 25 = 35x
→ 280 = 35x
→ x = 280 + 35 = 8
Therefore,
Length of the park = 5x = 5 x 8 = 40 m
Breadth of the park = 2x = 2 x 8 = 16 m
Question: 16
A square lawn is surrounded by a path 2.5 m wide. If the area of the path is 165 m2 , find the area of the lawn.
Solution:
Let the side of the lawn be x m.
Given that width of the path = 2.5 m
Side of the lawn including the path = (x + 2.5 + 2.5) m = (x + 5 ) m
So, area of lawn = (Area of the lawn including the path) – (Area of the path)
We know that the area of a square = (Side)2
Area of lawn (x2) = (x + 5)2 – 165
→ x2 = (X2 + 10X + 25) – 165
→ 165 = 10x + 25
→ 165 – 25 =10x
→ 140 = 10x
Therefore x = 140 / 10 = 14
Thus the side of the lawn = 14 m
Hence,
The area of the lawn = (14 m) 2 = 196 m2
Exercise 20.3
Question: 1
Find the area of a parallelogram with base 8 cm and altitude 4.5 cm.
Solution:
We have,
Base = 8 cm and altitude = 4.5 cm
Thus, Area of the parallelogram = Base x Altitude
= 8 cm x 4.5 cm
= 36 cm2
Question: 2
Find the area in square metres of the parallelogram whose base and altitudes are as under
(i) Base =15 dm, altitude = 6.4 dm
(ii) Base =1 m 40 cm, altitude = 60 cm
Solution:
We have,
(i) Base = 15 dm = (15 x 10) cm = 150 cm = 1.5 m
Altitude = 6.4 dm = (6.4 x 10) cm = 64 cm = 0.64 m
Thus, Area of the parallelogram = Base x Altitude
= 1.5 m x 0.64 m
= 0.96 m2
(ii) Base = 1 m 40 cm = 1.4 m [Since 100 cm = 1 m]
Altitude = 60 cm = 0.6 m
Thus, Area of the parallelogram = Base x Altitude
= 1.4 m x 0.6 m
= 0.84 m2 [Since 100 cm = 1 m]
Question: 3
Find the altitude of a parallelogram whose area is 54 d m2 and base is 12 dm.
Solution:
We have,
Area of the given parallelogram = 54 d m2
Base of the given parallelogram = 12 dm
Altitude of the given parallelogram = Area/Base = 54/12 dm = 4.5 dm
Question: 4
The area of a rhombus is 28 m2. If its perimeter be 28 m, find its altitude.
Solution:
We have,
Perimeter of a rhombus = 28 m 4(Side) = 28 m [Since perimeter = 4(Side)]
Side = 28 m4=7 m
Now, Area of the rhombus = 28 m2
(Side x Altitude) = 28 m2 (7 m x Altitude) = 28 m2
Altitude = 28 m 27m = 4 m
Question: 5
In Fig., ABCD is a parallelogram, DL ⊥ AB and DM ⊥ BC. If AB = 18 cm, BC =12 cm and DM= 9.3 cm, find DL.
Solution:
We have,
Taking BC as the base, BC = 12 cm and altitude DM = 9.3 cm
Area of parallelogram ABCD = Base x Altitude = (12 cm x 9.3 cm) = 111.6 c m2 —–(i)
Now, Taking AB as the base,
We have, Area of the parallelogram ABCD = Base x Altitude
= (18 cm x DL) —–(ii)
From (i) and (ii), we have 18 cm x DL = 111.6 c m2
DL = 111.6 cm 218 cm = 6.2 cm
Question: 6
The longer side of a parallelogram is 54 cm and the corresponding altitude is 16 cm. If the altitude corresponding to the shorter side is 24 cm, find the length of the shorter side.
Solution:
We have,
ABCD is a parallelogram with the longer side AB = 54 cm and corresponding altitude AE = 16 cm. The shorter side is BC and the corresponding altitude is CF = 24 cm.
Area of a parallelogram = base x height.
We have two altitudes and two corresponding bases.
So,
½ x BC x CF = ½ x AB x AE
= BC x CF = AB x AE
= BC x 24 = 54 x 16
= BC = (54×16)/24 = 36 cm
Hence, the length of the shorter side BC = AD = 36 cm.
Question: 7
In Fig. 21, ABCD is a parallelogram, DL ⊥ AB. If AB = 20 cm, AD = 13 cm and area of the parallelogram is 100 c m2, find AL.
Solution:
We have,
ABCD is a parallelogram with base AB = 20 cm and corresponding altitude DL.
It is given that the area of the parallelogram ABCD = 100 c m2
Now, Area of a parallelogram = Base x Height
100 c m2 = AB x DL
100 c m2 = 20 cm x DL
DL = 100 c m2= 5 cm
Again by Pythagoras theorem, we have,
(AD)2 = (AL)2 + (DL)2
= (13)2 = (AL)2+ (5)2
(AL)2= (13)2 – (5)2
= 169 – 25 = 144
(AL)2 = (12)2
AL = 12 cm
Hence, length of AL is 12 cm.
Question: 8
In Fig. 21, if AB = 35 cm, AD= 20 cm and area of the parallelogram is 560 cm2, find LB.
Solution:
We have,
ABCD is a parallelogram with base AB = 35 cm and corresponding altitude DL.
The adjacent side of the parallelogram AD = 20 cm.
It is given that the area of the parallelogram ABCD = 560 c m2
Now, Area of the parallelogram = Base x Height
560 cm2 = AB x DL 560 c m2 = 35 cm x DL
DL = 560 cm/235 cm= 16 cm
Again by Pythagoras theorem, we have, (AD)2 = (AL)2 + (DL)2
(20)2= (AL)2 + (16)2
(AL)2 = (20)2 – (16)2
= 400 – 256
= 144
(AL)2 = (12)2
= AL = 12 cm
From the figure, AB = AL + LB 35 cm
= 12 cm + LB
LB = 35 cm – 12 cm = 23 cm
Hence, length of LB is 23 cm.
Question: 9
The adjacent sides of a parallelogram are 10 m and 8 m. If the distance between the longer sides is 4 m, find the distance between the shorter sides.
Solution:
We have,
ABCD is a parallelogram with side AB = 10 m and corresponding altitude AE = 4 m.
The adjacent side AD = 8 m and the corresponding altitude is CF.
Area of a parallelogram = Base x Height
We have two altitudes and two corresponding bases.
So, AD x CF = AB x AE = 8 m x CF = 10 m x 4 m
= CF = (10 x 4)8 = 5 m
Hence, the distance between the shorter sides is 5 m.
Question: 10
The base of a parallelogram is twice its height. If the area of the parallelogram is 512 cm2, find the base and height.
Solution:
Let the height of the parallelogram be x cm.
Then the base of the parallelogram is 2x cm.
It is given that the area of the parallelogram = 512 cm2
So, Area of a parallelogram = Base x Height
512 c m2 = (2x) (x)
512 c m2 = 2x2
X2 = 512 cm2/2 = 256 c m2
X2 = (16 cm)2
X = 16 cm
Hence, base = 2x = 2 x 16 = 32 cm and height = x = 16 cm.
Question: 11
Find the area of a rhombus having each side equal to 15 cm and one of whose diagonals is 24 cm.
Solution:
Let ABCD be the rhombus where diagonals intersect at 0.
Then AB = 15 cm and AC = 24 cm.
The diagonals of a rhombus bisect each other at right angles.
Therefore, triangle A0B is a right-angled triangle, right angled at O such that
OA = ½(AC) = 12 cm and AB = 15 cm.
By Pythagoras theorem, we have,
(AB)2 = (OA)2 + (OB)2
(15)2 = (12)2 + (OB)2
(OB)2= (15)2– (12)2
(OB)2 = 225 – 144 = 81
(OB)2 = (9)2
OB = 9 cm
BD = 2 x OB = 2 x 9 cm = 18 cm
Hence, Area of the rhombus ABCD = (½ x AC x BD)
=( 1/2 x 24 x 18 )
= 216 cm2
Question: 12
Find the area of a rhombus, each side of which measures 20 cm and one of whose diagonals is 24 cm.
Solution:
Let ABCD be the rhombus whose diagonals intersect at 0.
Then AB = 20 cm and AC = 24 cm.
The diagonals of a rhombus bisect each other at right angles.
Therefore Triangle AOB is a right-angled triangle, right angled at O
Such that;
OA = ½ AC =12 cm and AB =20 cm
By Pythagoras theorem, we have,
(AB)2 = (OA)2+ (OB) 2
(20)2 = (12)2 + (OB) 2
(OB) 2= (20) 2– (12) 2
(OB)2 = 400 – 144
= 256
(OB) 2 = (16) 2
= OB = 16 cm
BD =2 x OB = 2 x 16 cm = 32 cm
Hence, Area of the rhombus ABCD = ½ x AC x BD
= 1/2 x 24 x 32
= 384 c m2
Question: 13
The length of a side of a square field is 4 m. What will be the altitude of the rhombus, if the area of the rhombus is equal to the square field and one of its diagonals is 2 m?
Solution:
We have,
Area of the rhombus = Area of the square of side 4 m
= 1/2 x AC x 130 = 4 m2
= 1/2 x AC x 2 m =16 m2
= AC = 16 m
We know that the diagonals of a rhombus are perpendicular bisectors of each other.
= AO = 1/2 (AC) = 8 m and BO = 1/2 (BD) = 1 m
By Pythagoras theorem, we have:
AO2 + BO2 = AB2
AB2 = (8 m) 2 + (1 m) 2 = 64 m2 + 1 m2 = 65 m2
Side of a rhombus = AB = √65 m.
Let DX be the altitude.
Area of the rhombus = AB x DX 16 m2
= √65m × DX
DX = 16/(√65) m
Hence, the altitude of the rhombus will be 16/√65 m.
Question: 14
Two sides of a parallelogram are 20 cm and 25 cm. If the altitude corresponding to the sides of length 25 cm is 10 cm, find the altitude corresponding to the other pair of sides.
Solution:
We have,
ABCD is a parallelogram with longer side AB = 25 cm and altitude AE = 10 cm.
As ABCD is a parallelogram. Hence AB = CD (opposite sides of parallelogram are equal).
The shorter side is AD = 20 cm and the corresponding altitude is CF.
Area of a parallelogram = Base x Height
We have two altitudes and two corresponding bases.
So, = AD x CF = CD x AE
= 20 x CF = 25 x 10
CF = 12.5 cm
Hence, the altitude corresponding to the other pair of the side AD is 12.5 cm.
Question: 15
The base and corresponding altitude of a parallelogram are 10 cm and 12 cm respectively. If the other altitude is 8 cm, find the length of the other pair of parallel sides.
Solution:
We have,
ABCD is a parallelogram with side AB = CD = 10 cm (Opposite sides of parallelogram are equal) and corresponding altitude AM = 12 cm. The other side is AD and the corresponding altitude is CN = 8 cm.
Area of a parallelogram = Base x Height
We have two altitudes and two corresponding bases.
So,
= AD x CN = CD x AM
= AD x 8 = 10 x 12
= AD = (10×12)/8=15 cm
Hence, the length of the other pair of the parallel side = 15 cm.
Question: 16
A floral design on the floor of a building consists of 280 tiles. Each tile is in the shape of a parallelogram of altitude 3 cm and base 5 cm. Find the cost of polishing the design at the rate of 50 paise per cm2.
Solution:
We have,
Attitude of a tile = 3 cm
Base of a tile = 5 cm
Area of one tile = Attitude x Base = 5 cm x 3 cm = 15 c m2
Area of 280 tiles = 280 x 15 c m2 = 4200 c m2
Rate of polishing the tiles at 50 paise per c m2 = Rs. 0.5 per c m2
Thus, Total cost of polishing the design = Rs. (4200 x 0.5) = Rs. 2100
Exercise 20.4
Question: 1
Find the area in square centimeters of a triangle whose base and altitude are as under :
(i) base =18 cm, altitude = 3.5 cm
(ii) base = 8 dm, altitude =15 cm
Solution:
We know that the area of a triangle = 1/2 (Base x Height)
(i) Here, base = 18 cm and height = 3.5 cm
Area of the triangle = 1/2 x 18 x 3.5
= 31.5 cm2
(ii) Here, base = 8 dm = (8 x 10) cm = 80 cm [Since 1 dm = 10 cm] and height = 3.5 cm
Area of the triangle = 1/2 x 80 x 15
= 600 c m2
Question: 2
Find the altitude of a triangle whose area is 42 cm2 and base is 12 cm.
Solution:
We have,
Attitude of a triangle = (2 x Area)/Base
Here, base = 12 cm and area = 42 cm2
Attitude = (2 x 42)/12 = 7 cm
Question: 3
The area of a triangle is 50 cm2. If the altitude is 8 cm, what is its base?
Solution:
We have,
Base of a triangle = (2 x Area)/ Altitude
Here, altitude = 8 cm and area = 50 cm2
Altitude = (2 x 50)/ 8 = 12.5 cm
Question: 4
Find the area of a right angled triangle whose sides containing the right angle are of lengths 20.8 m and 14.7 m.
Solution:
In a right-angled triangle,
The sides containing the right angles are of lengths 20.8 m and 14.7 m.
Let the base be 20.8 m and the height be 141 m.
Then,
Area of a triangle = 1/2 (Base x Height)
= 1/2 (20.8 × 14.7)
= 152.88 m2
Question: 5
The area of a triangle, whose base and the corresponding altitude are 15 cm and 7 cm, is equal to area of a right triangle whose one of the sides containing the right angle is 10.5 cm. Find the other side of this triangle.
Solution:
For the first triangle, we have,
Base = 15 cm and altitude = 7 cm
Thus, area of a triangle = 1/2 (Base x Altitude)
= 1/2 (15 x 7)
= 52.5 cm2
It is given that the area of the first triangle and the second triangle are equal.
Area of the second triangle = 52.5 c m2
One side of the second triangle = 10.5 cm
Therefore, The other side of the second triangle = (2 x Area)/One side of a triangle
= (2x 52.5)/10.5
=10 cm
Hence, the other side of the second triangle will be 10 cm.
Question: 6
A rectangular field is 48 m long and 20 m wide. How many right triangular flower beds, whose sides containing the right angle measure 12 m and 5 m can be laid in this field?
Solution:
We have,
Length of the rectangular field = 48 m
Breadth of the rectangular field = 20 m
Area of the rectangular field = Length x Breadth = 48 m x 20 m = 960 m2
Area of one right triangular flower bed = ½ (12 m x 5m) = 30 m2
Therefore,
Required number of right triangular flower beds = 960 m2/30 m2= 32
Question: 7
In Figure, ABCD is a quadrilateral in which diagonal AC = 84 cm; DL ⊥AC, BM ⊥ AC, DL = 16.5 cm and BM = 12 cm. Find the area of quadrilateral ABCD.
Solution:
We have,
AC = 84 cm, DL = 16.5 cm and BM = 12 cm
Area of triangle ADC = 1/2 (AC x DL) = 1/2 (84 cm x 16.5 cm) = 693 cm2
Area of triangle ABC = 1/2 (AC x BM) = 1/2 (84 cm x 12 cm) = 504 cm2
Hence, Area of quadrilateral ABCD = Area of ADC + Area of ABC = (693 + 504) cm2 = 1197 cm2
Question: 8
Find the area of the quadrilateral ABCD given in Figure. The diagonals AC and BD measure 48 m and 32 m respectively and are perpendicular to each other.
Solution:
We have,
Diagonal AC = 48 cm and diagonal BD = 32 m
Area of a quadrilateral = 1/2 (Product of diagonals)
= 1/2(AC x BD) = 1/2 (48 x 32) m2
= (24 x 32) m2 = 768 m2
Question: 9
In Fig below, ABCD is a rectangle with dimensions 32 m by 18 m. ADE is a triangle such that EF⊥ AD and EF= 14 cm. Calculate the area of the shaded region.
Solution:
We have,
Area of the rectangle = AB x BC = 32 m x 18 m = 576 m2
Area of the triangle = 1/2 (AD x FE)
= 1/2 (BC x FE) [Since AD = BC]
= 1/2 (18 m x 14 m)
= 9 m x 14 m
= 126 m2
Area of the shaded region = Area of the rectangle – Area of the triangle
= (576 – 126) m2
= 450 m2
Question: 10
In Fig. below, ABCD is a rectangle of length AB = 40 cm and breadth BC = 25 cm. If P, Q, R, S be the mid-points of the sides AB, BC, CD and DA respectively, find the area of the shaded region.
Solution:
Join points PR and SQ. These two lines bisect each other at point 0.
Here, AB = DC = SQ = 40 cm and AD = BC = RP = 25 cm
Also OP = OR = RP/2 = 25/2 = 12.5 cm
From the figure we observed that,
Area of Triangle SPQ = Area of Triangle SRQ
Hence, area of the shaded region = 2 x (Area of SPQ)
= 2 x (1/2 (SQ x OP))
= 2 x (1/2 (40 cm x 12.5 cm))
= 500 cm2
Question: 11
Calculate the area of the quadrilateral ABCD as shown in Figure, given that BD = 42 cm, AC = 28 cm, OD = 12 cm and AC ⊥ BO.
Solution:
We have,
BD = 42 cm, AC = 28 cm, OD= 12 cm
Area of Triangle ABC = 1/2 (AC x OB)
= 1/2 (AC x (BD – OD))
= 1/2 (28 cm x (42 cm – 12 cm))
= 1/2 (28 cm x 30 cm)
= 14 cm x 30 cm
= 420 cm2
Area of Triangle ADC = 1/2 (AC x OD) = 1/2 (28 cm x 12 cm)
= 14 cm x 12 cm
= 168 cm2
Hence, Area of the quadrilateral ABCD = Area of ABC + Area of ADC
= (420 + 168) cm2
= 588 cm2
Question: 12
Find the area of a figure formed by a square of side 8 cm and an isosceles triangle with base as one side of the square and perimeter as 18 cm.
Solution:
Let x cm be one of the equal sides of an isosceles triangle.
Given that the perimeter of the isosceles triangle = 18 cm
Then, x + x + 8 =18
2x = (18 – 8) cm = 10 cm
x = 5 cm
Area of the figure formed = Area of the square + Area of the isosceles triangle
Question: 13
Find the area of Figure, in the following ways: (i) Sum of the areas of three triangles (ii) Area of a rectangle — sum of the areas of five triangles
Solution:
We have,
(i) P is the midpoint of AD.
Thus AP = PD = 25 cm and AB = CD = 20 cm
From the figure, we observed that,
Area of Triangle APB = Area of Triangle PDC
Area of Triangle APB = 1/2 (AB x AP ) = 1/2 (20 cm x 25 cm) = 250 cm2
Area of Triangle PDC = Area of Triangle APB = 250 c m2
Area of Triangle RPQ = 1/2 (Base x Height) = 1/2 (25 cm x 10 cm) = 125 cm2
Hence, Sum of the three triangles = (250 + 250 + 125) cm2 = 625 cm2
(ii) Area of the rectangle ABCD = 50 cm x 20 cm = 1000 cm2
Thus, Area of the rectangle – Sum of the areas of three triangles
= (1000 – 625 ) cm2 = 375 cm2
Question: 14
Calculate the area of quadrilateral field ABCD as shown in Figure, by dividing it into a rectangle and a triangle.
Solution:
We have,
Join CE , which intersect AD at point E.
Here, AE = ED = BC = 25 m and EC = AB = 30 m
Area of the rectangle ABCE = AB x BC = 30 m x 25 m = 750 m2
Area of Triangle CED = 1/2 (EC x ED) = 1/2 ( 30 m x 25 m) = 375 m2
Hence, Area of the quadrilateral ABCD = (750 + 375) m2 = 1125 m2
Question: 15
Calculate the area of the pentagon ABCDE, where AB = AE and with dimensions as shown in Figure.
Solution:
Join BE.
Area of the rectangle BCDE = CD x DE
= 10 cm x 12 cm
= 120 c m2
Area of Triangle ABE = 1/2 (BE x height of the triangle)
= 1/2 (10 cm x (20 – 12) cm)
= 1/2 (10 cm x 8 cm)
= 40 cm2
Hence, Area of the pentagon ABCDE = (120 + 40) cm2 = 160 cm2
Question: 16
The base of a triangular field is three times its altitude. If the cost of cultivating the field at Rs 24.60 per hectare is Rs 332.10, find its base and height.
Solution:
Let altitude of the triangular field be h m
Then base of the triangular field is 3h m.
Area of the triangular field = 1/2 (h x 3h )=3h2/2 m2 —–(i)
The rate of cultivating the field is Rs 24.60 per hectare.
Therefore,
Area of the triangular field = 332.10 /24.60
= 13.5 hectare = 135000 m2 [Since 1 hectare = 10000 m2] —–(ii)
From equation (i) and (ii) we have,
3h2/2 = 135000 m2
3h2 = 135000 x 2 = 270000 m2
h2 = 270000/3 m2= 90000 m2 = (300 m)2
h = 300 m
Hence, Height of the triangular field = 300 m and base of the triangular field = 3 x 300 m = 900 m
Question: 17
A wall is 4.5 m long and 3 m high. It has two equal windows, each having form and dimensions as shown in Figure. below. Find the cost of painting the wall (leaving windows) at the rate of Rs 15 per m2.
Solution:
We have,
Length of a wall = 4.5 m
Breadth of the wall = 3 m
Area of the wall = Length x Breadth
= 4.5 m x 3 m = 13.5 m2
From the figure we observed that,
Area of the window = Area of the rectangle + Area of the triangle
= (0.8 m x 0.5 m) + (12 x 0.8 m x 0.2 m) [Since 1 m = 100 cm]
= 0.4 m2 + 0.08 m2
= 0.48 m2
Area of two windows = 2 x 0.48 = 0.96 m2
Area of the remaining wall (leaving windows) = (13.5 – 0.96) m2
= 12.54 m2
Cost of painting the wall per m2 = Rs. 15
Hence, the cost of painting on the wall = Rs. (15 x 12.54)
= Rs. 188.1
(In the book, the answer is given for one window, but we have 2 windows.)
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